If `tan ^(-1) ((1-x)/(1+x))= (1)/(2) tan^(-1) x, x gt 0`, then x = ?

Options

A. `(1)/sqrt(3)`
B. `-(1)/sqrt(3)`
C. `-sqrt(3)`
D. None of these

Answer

To solve the equation \( \tan^{-1} \left( \frac{1-x}{1+x} \right) = \frac{1}{2} \tan^{-1} x \) for \( x > 0 \), we can follow these steps:

### Step 1: Use the formula for the tangent of the difference of angles
We can use the identity for the tangent of the difference of two angles:
\[ \tan^{-1} a - \tan^{-1} b = \tan^{-1} \left( \frac{a - b}{1 + ab} \right) \]
In this case, we can rewrite the left side:
\[ \tan^{-1} 1 - \tan^{-1} x = \tan^{-1} \left( \frac{1 - x}{1 + x} \right) \]
Thus, we have:
\[ \tan^{-1} 1 - \tan^{-1} x = \frac{1}{2} \tan^{-1} x \]

### Step 2: Substitute \(\tan^{-1} 1\)
We know that \( \tan^{-1} 1 = \frac{\pi}{4} \). Therefore, we can rewrite the equation as:
\[ \frac{\pi}{4} - \tan^{-1} x = \frac{1}{2} \tan^{-1} x \]

### Step 3: Combine the terms
Now, we can combine the terms involving \( \tan^{-1} x \):
\[ \frac{\pi}{4} = \tan^{-1} x + \frac{1}{2} \tan^{-1} x \]
This simplifies to:
\[ \frac{\pi}{4} = \frac{3}{2} \tan^{-1} x \]

### Step 4: Solve for \(\tan^{-1} x\)
To isolate \( \tan^{-1} x \), we multiply both sides by \( \frac{2}{3} \):
\[ \tan^{-1} x = \frac{2}{3} \cdot \frac{\pi}{4} = \frac{\pi}{6} \]

### Step 5: Take the tangent of both sides
Now we take the tangent of both sides:
\[ x = \tan\left(\frac{\pi}{6}\right) \]

### Step 6: Calculate \(\tan\left(\frac{\pi}{6}\right)\)
We know that:
\[ \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \]
Thus, we find:
\[ x = \frac{1}{\sqrt{3}} \]

### Conclusion
The value of \( x \) is \( \frac{1}{\sqrt{3}} \).